OpenMPT has a new built-in plugin: It's called LFO and that's exactly what it is. This is the answer to all those plugins whose LFOs lack certain waveforms, cannot control certain parameters or don't have an LFO to begin with. The LFO plugin sends its automation data through the plugin chain that you should already be familiar with, i.e. you simply select the target of the LFO plugin through the "output to" dropdown in the General Tab or in the plugin GUI.

You can control plugin parameters as well as MIDI CCs on any channel. The phase of the plugin can be set manually by automating parameter
7 8, if required. This way, you can reset the phase whenever you like, but not only to the beggining of the phase but to any position you like.
The plugin is not finalized yet, so parameter ranges and other things might still change before the release of OpenMPT 1.27. I have hardly used the plugin myself yet so I'm looking for ideas what can be improved to make it more useful or easier to use.
Grab a recent test build from
https://builds.openmpt.org/builds/ and start testing. Report your findings here!